Assignment 7

Due in class on Thursday, October 31

Problem 1. Assuming that the following function is periodically extended, find its representation in a Fourier series

$f(x) = \displaystyle \left\{\begin{array}{lcr} \pi+x\;,&\quad&-\pi \le x < 0 \\ \pi-x\;,&\quad& 0 \le x < \pi\end{array}\right.$

In [10]:

show(plot(pi+x,(x,-pi,0))+plot(pi-x,(x,0,pi)))

Problem 2. The Fourier transform is defined as

$\displaystyle F(\omega)=\int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dt$

with the inverse transform given by

$\displaystyle f(t)=\frac{1}{2\pi}\,\int_{-\infty}^{\infty}F(\omega)e^{i\omega t}\,d\omega$ ;.

The Ricker wavelet, also known as the Mexican hat wavelet, is a popular representation of seismic signals. The Ricker wavelet $r(t)$ is defined as the second derivate of the Gaussian

$r(t)=\displaystyle -\frac{d^2}{d t^2} g(t)$ , where $g(t)=e^{-a^2\,t^2}$ .

Find the Fourier transform of the Ricker wavelet and determine:

peak frequency: the frequency of the maximum in the Fourier spectrum;
centroid frequency: the "center of mass" for the positive part of the spectrum;
apparent frequency: the frequency corresponding to the period between two minima in the time domain.

In [8]:

t=var('t')
gauss(t)=exp(-t^2)
ricker(t)=-gauss.diff(t,2)
plot(ricker(t),t,-5,5,color='red',thickness=3)

In [0]: